We now investigate the charge injection and transport mechanisms in the SL and BL devices. Across a metal/semiconductor contact, charge injection occurs via three distinct pathways (see Figure 4a-c): (i) direct tunneling (DT) at low \(V_\mathrm {DS}\) ;[1]} (ii) Richardson-Schottky (RS) thermionic injection at moderate \(V_\mathrm {DS}\) ;[2]}, [3]} and (iii) field-induced Fowler-Nordheim (FN) tunneling at high \(V_\mathrm {DS}\)  [4]} where the current-voltage scaling relations are governed, respectively, by
\(\ln \left(\frac{I_\mathrm {DS}}{V_\mathrm {DS}^2}\right)\propto \ln \left(\frac{1}{V_\mathrm {DS}} \right) \& \text{ , DT}\) 
\(\ln \left(I_\mathrm {DS}\right) \propto V_\mathrm {DS}^{1/2} \& \text{ , RS emission}\) 
\(\ln \left(\frac{I_\mathrm {DS}}{V_\mathrm {DS}^2}\right) \propto -\frac{1}{V_\mathrm {DS}} \& \text{ , FN tunneling}\) 
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